## Chebyshevs Inequality

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# Chebyshev’s Inequality

Chebyshev’s inequality gives the minimum value of proportion that holds true for any sample data distribution a certain number of standard deviations from the mean. Hence the true proportion can either be equal to or greater than the result obtained from Chebyshev’s inequality.

Chebyshev’s inequality tells us that for any sample data, at least 1-1/K^{2} of it will be K standard deviations from the mean.

**Conditions for K**

For Chebyshev’s inequality to hold, the following conditions of the value of K need to be satisfied:

- The value of K needs to be greater than zero i.e. K > 0.
- K needs to be a real number.

**Case Scenario 1**

From the empirical rule, we know that approximately 95 % of sample data that is normally distributed will lie two standard deviations from the mean.

Substituting for K = 2, Chebyshev’s inequality evaluates as follows;

1-1/2^{2}=3/4

*We find that at least 75 % of data values of any distribution will lie one standard deviation from the mean.*

**Case Scenario 2**

From the empirical rule, we know that approximately 99.7 % of sample data that is normally distributed will lie three standard deviations from the mean.

Let’s find out using Chebyshev’s inequality at least how many data values will lie three standard deviations from the mean of any distribution.

For K = 3, it follows that;

1-1/3^{2}=8/9

*We find that at least 88.89 % of data values of any distribution will lie one standard deviation from the mean.*